# Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

## TL;DR

This paper provides a geometric characterization of the conditions under which harmonic measure is absolutely continuous with respect to surface measure, linking it to the solvability of the $L^p$ Dirichlet problem for domains with Ahlfors-David regular boundaries.

## Contribution

It offers a geometric criterion for the weak-$A_
Infty$ property of harmonic measure, connecting it to $L^p$ solvability of the Dirichlet problem under natural geometric conditions.

## Key findings

- Characterizes weak-$A_
Infty$ property geometrically.
- Establishes equivalence between harmonic measure properties and $L^p$ solvability.
- Provides counterexamples showing sharpness of conditions.

## Abstract

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ \Omega\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $\Omega$, with data in $L^p(\partial\Omega)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds); moreover, the examples show that the upper and lower Ahlfors-David bounds are each quantitatively sharp.